Numerical Examples

Up to now, this chapter has derived two unconstrained minimization formulations (3.28) and (3.29) for the SCNE models. Compared to the modified projection method, it has qualitatively shown that the quasi-Newton algorithm is more suitable to solve the SCNE models based the unconstrained minimization formulations. To evaluate the performances of these two solution methods, benchmark examples are essential. To do so, eleven examples for the SCNE models will be employed. The first benchmark example that will be constructed is a variation of Example 1 in Nagurney et al. (2002).

The other ten examples those will be utilized are the same as that given by Nagurney et al. (2002) and Dong et al. (2004).

To compare the two solution methods fairly, the same initial solutions and the same stopping criterion in them for these eleven examples are adopted, which are shown as follows.

Five benchmark examples with deterministic demands:

( )

Initial solution: X(0) = 10,10, ,10 ∈ℜ^{mn no n o+ + +} (3.41)

( 1) ( )

Stopping criterion: X ^N+ −X ^N ≤0.0001 (3.42) Six benchmark examples with random demands:

( )

quasi-Newton algorithm in the optimization tool box of the Matlab is invoked directly.

These two solution methods are run on a personal computer with the CPU of Intel Pentium IV 1.6GMHZ and RAM of 256M.

3.5.1 A modified example

Let us consult Example 1 for the SCNE model with deterministic demands, given by Nagurney et al. (2002). Based on this example, a new numerical example is constructed as follows. It keeps all data in the original example except for the production cost functions, which now take the new expressions:

3

Applying the quasi-Newton algorithm for the relevant constrained minimization model (3.28) with respect to the modified example yields the solution:

( )

Figure 3.3 illustrates the change of value of the merit function within the last

fifteen iterations of the quasi-Newton algorithm in solving the modified example. It clearly indicates that the value of the merit function at the iterative point is almost equal to zero after 61 iterations. In other words, the above solutionX is indeed the ^* solution of this example according to Proposition 3.1.

0

Figure 3.3 Change of value of merit function with respect to the number of iterations for the modified example

Since the second-order derivatives for production cost functions (3.45) and (3.46) are unbounded, Lipschitz continuity condition for the vector function ^{F X}

( )

associated with this example does not hold. It thus means that the modified projection method may not be convergent for the example according to Theorem 4 of Nagurney et al. (2002). However, by trial and error on step size α in iterative scheme (3.38), it is found that find that when step size α in the effective interval (0, 0.005] the

the modified projection method with three different predetermined step sizes for the example. It can be seen that the modified projection method is divergent in the case of that step size α =0.01. In addition, the modified projection method will reach the stopping criterion (3.42) after 2315 iterations when step size α =0.005, but it will terminate after 9820 iterations when step size α =0.001. Hence, the performance of the modified projection method heavily depends on the value of the predetermined step size. Unfortunately, it is not an easy task to seek an appropriate step size for a SCNE problem. Distance between two consective iterative points

Step size =0.01 Step size = 0.005 Step size =0.001

Figure 3.4 The convergent performance of the modified projection method

In terms of CPU times used by these two solution methods for the modified example, the quasi-Newton algorithm has spent 0.37 seconds, and the modified projection method with step size ˆα =0.05 has used 0.73 seconds. While both of these two numbers are acceptable in finding a solution, they explicitly imply that performance of the quasi-Newton method for this example is better than that of the

modified projection method.

3.5.2 The other ten examples

Nagurney et al. (2002) provided four examples about the supply network equilibrium model with deterministic demands, and Dong et al. (2004) gave six examples for the supply network equilibrium model with random demands. They merely employed these ten examples to verify that the modified projection method is capable of solving their variational inequality formulations (3.7) and (3.14). The predetermined step sizes guaranteeing the convergence of the modified projection method expressed in eqns. (3.38)-(3.39) for these ten examples do theoretically exist.

However, there is no practical guide to obtain these step sizes. By trial and error, an effective interval of the step size for each example can be estimated, which is tabulated in Tables 3.1 and 3.2, respectively. According to these two tables, it can be seen that the effective interval of the step size is varied over the ten examples.

With regard to these ten examples, both the quasi-Newton algorithm and the modified projection method with a step size in the corresponding effective interval shown in Tables 3.1 and 3.2 can generate the same solution. The CPU times they used for each example, however, are quite different. They can be compared by calculating ratio of the CPU time used by the quasi-Newton algorithm to the least CPU time used by the modified projection method, which are obtained by enumerating all possible effective step sizes. These ratios are listed in Tables 3.3 and 3.4. From these two tables, it can be seen that there are 6 cases out of the ten examples, among which the

performance of the quasi-Newton algorithm is better than the modified projection method. If the predetermined step size in the modified projection method is equal to 0.01, the number of examples for which the CPU time used by the quasi-Newton is less than that used by the modified projection method will rise to 8.

Table 3.1 Effective intervals of step size α for the four examples in Nagurney et al.

(2002)

Example 1 Example 2 Example 3 Example 4 (0, 0.06] (0, 0.06] (0, 0.04] (0, 0.06]

Table 3.2 Effective intervals of step size ˆα for the six examples in Dong et al. (2004) Example 1 Example 2 Example 3 Example 4 Example 5 Example 6

(0, 0.01] (0, 0.02] (0, 0.03] (0, 0.03] (0, 0.03] (0, 0.02]

Table 3.3 Ratios of CPU time in seconds used by the quasi-Newton algorithm to the least CPU time used by the modified projection method for the four examples of

Nagurney et al. (2004)

Example 1 Example 2 Example 3 Example 4 11.90 2.16 62.36 4.38

Table 3.4 Ratios of CPU time in seconds used by the quasi-Newton algorithm to the least CPU time used by the modified projection method for the six examples of Dong

et al. (2004)

Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 0.01 0.16 0.23 0.54 0.51 0.17